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```ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1

Development and Simulation of Mathematical
Modelling of Hydraulic Turbine

Gagan Singh and D.S. Chauhan
^ttarakhand Technical University, Dehradun, (U.K), India.

hod_ee@dit.edu.in
2 Uttarakhand Technical University, Dehradun, (U.K),India.

pdschauhan @ gmail . com

Abstract- Power system performance is affected by dynamic
characteristics of hydraulic governor-turbines during and
following any disturbance, such as occurrence of a fault,
loss of a transmission line or a rapid change of load. Accurate
modelling of hydraulic System is essential to characterize
mathematical modeling of hydraulic turbine is presented. The
model is capable to implement the digital systems for
monitoring and control replacing the conventional control
systems for power, frequency and voltage. This paper presents
the possibilities of modeling and simulation of the hydro power
plants and performs an analysis of different control structures
and algorithms.

Key words: mathematical modeling, simulation, hydraulic
turbine.

I. INTRODUCTION

Usually, a typical hydroelectric power plant consists of
water tunnel, penstock, surge tank, hydraulic turbine, speed
governor, generator and electric network. Hydrodynamics
and mechanoelectric dynamics are all involved in such a
nonlinear dynamic system. Generally, a hydroelectric
generating unit has many different operating conditions and
change in any operating condition results in small or large
hydraulic transients. There are many instances of damage to
penstock or hydraulic turbine which are most probably
occurring due to large transients. Hydrodynamics is
influenced by the performance of hydraulic turbine which
depends on the characteristics of the water column feeding
the turbine. These characteristics include water inertia, water
compressibility and pipe wall elasticity in the penstock.
Different construction of hydropower systems and different
operating principles of hydraulic turbines make difficult to
develop mathematical models for dynamic regime, in order to
design the automatic control systems. Also, there are major
differences in the structure of these models. Moreover, there
are major differences due to the storage capacity of the
reservoir and the water supply system from the reservoir to
the turbine (with or without surge chamber). The dynamic
model of the plants with penstock and surge chamber is more
complicated than the run-of-the-river plants, since the water
feed system is a distributed parameters system. This paper
will present several possibilities for the modeling of the
hydraulic systems and the design of the control system.

II. MODELING OF THE HYDRAULIC SYSTEM

For run-of-the river types of hydropower plants have a

DOL01.UCSI.02.02.1

low water storage capacity in the reservoir; therefore the
plant operation requires a permanent balance between the
water flow through turbines and the river

i

4*uflbl -J

ftffauifi

I

*BHw

Hfr -

411

!■]

Figure 1: Functional block diagram of hydraulic governor-turbine
system interconnected with a power system network.

flow in order to maximize the water level in the reservoir for a
maximum efficiency of water use. Next, we will determine the
mathematical model for each component of the hydropower
system.

A. Mathematical modeling of Hydraulic turbine

The representation of the hydraulic turbine and water
column in stability studies is usually based on the following
assumptions:-

> The hydraulic resistance is negligible.

> The penstock pipe is inelastic and the water is
incompressible.

> The velocity of the water varies directly with the
gate opening and with the square root of the net

> The turbine output power is proportional to the
product of head and volume flow.

The hydraulic turbine can be considered as an element
without memory since the time constants of the turbine are
less smaller than the time constants of the reservoir, penstock,
and surge chamber, if exists, which are series connected
elements in the system. As parameters describing the mass
transfer and energy transfer in the turbine we will consider
the water flow through the turbine Q and the moment M
generated by the turbine and that is transmitted to the
electrical generator. These variables can be expressed as non-
linear functions of the turbine rotational speed N, the turbine
gate position Z, and the net head H of the hydro system.
Q = Q(H,N,Z) (1)

M = M(H,N,Z) (2)

Through linearization of the equations (1) and (2) around
the steady state values, we obtain:

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ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1

&Q = 3 JL &H + *R &N + 3 JL AZ

^ 3H 3N 3z

q= ajj A+ fljj w+ fljj 2

3H 3jY 3Z

fl2= flj j A+ iJjJ JT+ iJjJ 2

(3)

(4)

Where the following notations were used:

_ — _ - V _ * M r. _ iJV **

^ — — , ?1 — , TYl — , fl — , Z — —

Qti iV[] Mg Hj^ Z[]

Which represent the non-dimensional variations of the
parameters around the steady state values.

B. The hydraulic feed system

The hydraulic feed system has a complex geometrical
configuration, consisting of pipes or canals with different
shapes and cross-sections. Therefore, the feed system will
be considered as a pipe with a constant cross-section and
the length equal with real length of the studied system. In
order to consider this, it is necessary that the real system and
the equivalent system to contain the same water mass. Let
consider m , m 2 .. .m n the water masses in the pipe zones having
the lengths L,L,...,l and cross- sections A „ A^,...,A of the real

V 2' ' n V 2' ' n

feed system. The equivalent system will have the length
L=l 1 -\-l 2 -\-...+l n and cross-section A, conveniently chosen.
In this case, the mass conservation law in both systems will

Since the water can be considered incompressible, the flow
Q. through each pipe segment with cross-section A. is
identical and equal with the flow Q through the equivalent
pipe

Q=v.A=Qi=Yi.A^ fori=l : 2 : ... : n (6)

Where v is the water speed in the equivalent pipe, and v. is
the speed in each segment of the real pipe.
From the mass conservation law it results:

v — 1 = fr^L _ £k

A T.i-i'Ai Y.i-i'Ai

n

The dynamic pressure loss can be computed considering the
inertia force of the water exerted on the cross-section of the
pipe:

F; = — m. a = — L. A. p. a = —A — — (S)

g dt

Where L is the length of the penstock or the feed canal, A is
the cross-section of the penstock, g? is the specific gravity
of water (1000Kgf/m 3 ), a is the water acceleration in the
equivalent pipe, and g=9.81 m/s 2 is the gravitational
acceleration. The dynamic pressure loss can be expressed
as:

_ F[ _ £-L dv _ y-L Y.[j dQ

V — Li — LZl _ —

S I liJ.i dt

P)

Using non-dimensional variations, from (9) it results:

DOL01.UCSI.02.02.1

4tf d _ _pn_ pLElj d {\jJ
tf dn ~ ** dn ' E kr*i ' dt
Or in non-dimensional form:

(10)

** = -T w *

■S ^

Where T w is the integration constant of the hydropower
system and the variables have the following meaning:

* dD QU * dD I IlAL L J X J

It must be noted that this is a simplified method to compute
the hydraulic pressure loss, which can be used for run-of-the
river hydropower plants, with small water head. If an exact
value of the dynamic pressure is required, then the formulas
presented in [8], sub-chapter 8.4 "The calculation of hydro
energy potential" shall be used.
Using the Laplace transform in relation (1 1), it results:

h d (\$) = -sT w .qlj},tmd <yfc) = - — h d (s') (13)

Replacing (13) in (3) and (4) and doing some simple
calculations, we obtain:

hi'W) = -

7

vU) -

*fc>

(14)

(15)

The mechanical power generated by the turbine can be
calculated with the relation P-^.g.Q.H, which can be used to
obtain the linearized relations for variations of these values

Where fj? is the turbine efficiency, and g, Q, and H were
defined previously.

On the other hand, the mechanical power can be determined
using the relation P=Mu=2?z:M.N, which can be used to obtain
the linearized relations for variations of these values around

7i = p — m

(IS)

56

Where P D m w is the steady state power generated by
the turbine for a given steady state flow Q and a steady
state head H Q , and N is the steady state rotational speed.
Using these relations, the block diagram of the hydraulic
turbine, for small variation operation around the steady state
point, can be determined and is presented in Figure 2, where
the transfer functions for different modules are given by the
following relation:

*- W = i^ T A « ^/"/ ^ = TTTT^<

14-0. . T. j. J h - 14-CL. . T^.j'

(-^^-) (19)

*k ACEEE

ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1

For an ideal turbine, without losses, the coefficients a..

y

resulted from the partial derivatives in equations (12 - 16)
have the following values: a n =0.5; a 12 =a 13 =l; a 21 =1.5; a 23 =l.
In this case, the transfer functions in the block diagram are
given by the following relation:

Table I. Variations of the time constant of the hydro system

JU*> = :

-jU*> = :

-.ff^Cs) = —

H,J S y = SO—, fl^C) = (l,5 *£-)

"-w=( i -^y=s

D.5."T W S

(20)

(21)

(»>

III. SIMULATION RESULTS

Example. Let consider a hydroelectric power system with
the following parameters:
-Water flow (turbines): Q N =725 m 3 /s;
-Water level in the reservoir: H N =30 m;
-The equivalent cross-section of the penstock A=60m 2 ;
-Nominal power of the turbine P N =178MW;
-Turbine efficiency rp0.94;
-Nominal rotational speed of the turbine=
N=71.43rot/min;
-The length of the penstock l=SL=20m;

*Utf

*M

JUtf

HfcW

— J j — jiki — J

2 J *Utt| I

r'r.X, P

Fig. 2. The block diagram of the hydraulic turbine.

It shall be determined the variation of the time constant T w

w

for the hydro power system.

For the nominal regime, using relation (12), where Jl.=20m,

the time constant of the system is:

r _ "5

:;

3D 9..BL.6D

= 0,82s

(23)

Next there is a study of the variation of the time constant
due to the variation of the water flow through the turbine for
a constant water level in the reservoir, H=30m, as well as the
variation due to the variable water level in the reservoir for a
constant flow Q=725 m 3 /s. In table I, column 3 and figure 3 a)
are presented the values and the graphical variation of the
time constant T w for the variation of the water flow between

w

500 m 3 /s and 110 m 3 /s, for a constant water level in the
reservoir, H=30m. In table 1. column 4 and figure 3 b) are
presented the values and the graphical variation of the time
constant T w for the variation of the water level in

w

DOL01.UCSI.02.02.1

H

Q

Tn(H=3Qm)

Tw«^7:Smc-s

17

L 1 *x*5

I.25fi0fi5

1.445101

20

5NS.14

1,093155

1 ,i3 1 736

23

&39 r 25

Q;9:5Q5tf9

1,071075

2*

742.42

0.S403S3:

.9WS9

29

6G5„e:

0.753S99S

0849473

32

G03.2I

0.0832217

0,709835

35

551,51

0,G24tf598

0,703849

J8

~--r- r

r 575344C

0.G4S2S2

the reservoir, for a constant water flow, Q=725m 3 /s.

500 6W -00 WO WO 1000 !IW 1*00

T* [»]

1,1 1 T

hi

W [ni| 40

Figure 3. Variation of the integral time constant TW: a) by the
flow Q, b) by the water level H

It can be seen from the table or from the graphs that the
time constant changes more than 50% for the entire
operational range of the water flow through the turbine or if
the water level in the reservoir varies. These variations will
create huge problems during the design of the control system
for the turbine, and robust control algorithms are
recommended.

In figure 4 the block diagram of the turbine's power con-
trol system, is presented using a secondary feedback from
the rotational speed of the turbine. It can be seen from this
figure that a dead-zone element was inserted in series with
the rotational speed sensor in order to eliminate the feedback
for ±0.5% variation of the rotational speed around the syn-
chronous value. This oscillation has no significant influence
on the performance of the system but would have lead to

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ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 201 1

permanent perturbation of the command sent to the turbine
gate.

%E1

j- 1 ' 'LTidE

Figure 4. Block Diagram of the control system for hydraulic
turbines

The constants of the transfer functions had been
computed for a nominal regime T w =0.8s. The optimal
parameters for a PI controller are: K R =10, T=0.02s. The results
of the turbine simulation for different operational regimes are
presented in figure 5, for a control system using feedbacks
from the turbine power and rotational speed, with a dead-
zone on the rotational speed channel for ±0.5% variation of
the rotational speed around the synchronous value (a) Power
variation with 10% around nominal value, b) Rotational speed
variation for power control).

■M

Figure 5 Control structure with feedbacks from turbine power and
rotational speed:

DOL01.UCSI.02.02.1

In figure 6 the variations of the turbine power (graph a) and
rotational speed (graph b) for the control system a feedback
from the turbine power but no feedback from the rotational
speed are presented.

(14

Figure 6 Control structure with only power feedback

a) Power variation with 10% around nominal value

b) Rotational speed variation for power control

IV. CONCLUSIONS

The detailed mathematical modeling of hydraulic turbine
is vital to capture essential system dynamic behavior .The
possibility of implementation of digital systems for monitoring
and control for power, frequency and voltage in the cascade
hydro power plant was discussed. The simplified
mathematical models, capable to accurately describe dynamic
and stationary behavior of the hydro units a developed and
simulated. These aspects are compared with experimental
results. Finally, a practical example was used to illustrate the
design of controller and to study the system stability.

Acknowledgement

The authors would like to thank Professor S.P. Singh (LIT
Roorkee, India) for his continuous support and valuable
suggestions.

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ACEEE Int. J. on Control System and Instrumentation, Vol. 02, No. 02, June 2011

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